Free path lengths in quasicrystals

Abstract

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g. at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of the Penrose tiling. Our main result proves the existence of a limit distribution of the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner's measure classification.

References in corpus (3)

• Kinetic transport in the two-dimensional periodic Lorentz gas
• Power-law distributions for the free path length in Lorentz gases
• Free path lengths in quasi crystals

Cited by in corpus (7)

• Aperiodic order and spherical diffraction, I: Auto-correlation of model sets
• An Effective Ratner Equidistribution Result for ASL(2,R)
• Horizons and free path distributions in quasiperiodic Lorentz gases
• Directions in Orbits of Geometrically Finite Hyperbolic Subgroups
• The Lorentz Process with a Nearly Periodic Distribution of Scatterers
• Statistical properties of Lorentz gases on aperiodic tilings, Part 1
• On the free path length distribution for linear motion in an n-dimensional box